explore equivalent expressions · involving rational numbers think about what you know about the...

©Curriculum Associates, LLC Copying is not permitted. LESSON 15 Write Equivalent Expressions Involving Rational Numbers 309

Previously, you worked with equivalent expressions that have positive coefficients. In this lesson, you will work with equivalent expressions that have negative coefficients.

➤➤ Use what you know to try to solve the problem below.

In a certain video game, players earn bonus points after finishing a

level. The number of bonus points is based on how many seconds, t,

it takes the player to finish the level. The game uses the expression

25.5 1 500 1 0.9t 1 1 11 ·· 2 2 1.9t 2 to determine the number of bonus points.

Zahara finishes a level in 201 seconds. How many bonus points does she earn?

Explore Equivalent Expressions

LESSON 15 | SESSION 1

TRY IT Math Toolkit grid paper, sticky notes

Ask: What did you do first to determine how many bonus points Zahara earns?

Share: First, I . . . because . . .

DISCUSS IT

Learning Target SMP 1, SMP 2, SMP 3, SMP 4, SMP 5, SMP 6, SMP 7Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.

©Curriculum Associates, LLC Copying is not permitted.LESSON 15 Write Equivalent Expressions Involving Rational Numbers310


CONNECT IT

Look Back How many bonus points does Zahara earn? What is another expression the game could use to determine the number of bonus points she earns?

Look Ahead In the Try It, you evaluated an expression. Sometimes it can be helpful to find an equivalent expression to evaluate instead. One way to find an equivalent expression is to use the distributive property.

a. Sometimes you can use the distributive property to expand an expression. You can expand 23(5 2 x) into the equivalent expression 215 1 3x. Show how you can use the distributive property to expand 24(2a 1 2) to get an equivalent expression.

b. Sometimes you can use the distributive property to factor an expression. You can factor 3z 2 6 to get the equivalent expression 3(z 2 2). Show how you can use the distributive property to factor 2g 2 10 to get an equivalent expression.

c. Can you factor 22x 1 6 to get 22(x 2 3)? Explain.

Reflect Explain why the expressions 26(5 2 k) and 6k 2 30 are equivalent.

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Aiden says that equivalent expressions always have the same number of terms. Is Aiden correct? If he is, explain why. If he is not correct, give a counterexample.

2

Prepare for Writing Equivalent Expressions Involving Rational Numbers

Think about what you know about the terms of an expression. Fill in each box. Use words, numbers, and pictures. Show as many ideas as you can.

1

LESSON 15 | SESSION 1 Name:

In My Own Words

Examples

My Illustrations

Non-Examples

term


Jake cuts out five-pointed stars. They are different sizes, but they

all have the same shape. The side lengths within each star are the same.

To find the perimeter of each star, Jake uses the expression

12, 1 1 ·· 2 2 4, 2 8.5 1 2, 1 8, where , is the side length of the star.

a. Find the perimeter of a star with side length 6 inches. Show your work.

SOLUTION

b. Check your answer to problem 3a. Show your work.

3



➤➤ Read and try to solve the problem below.

Are 2 1 ·· 3 (23m 1 6 2 12 1 15m) and 2(1 2 2m) equivalent expressions?

Show why or why not.

Develop Expanding Expressions


TRY IT Math Toolkit grid paper, sticky notes

Ask: How did you get started figuring out whether the expressions are equivalent?

Share: I started by . . .

DISCUSS IT


➤➤ Explore different ways to find equivalent expressions.

Are 2 1 ·· 3 (23m 1 6 2 12 1 15m) and 2(1 2 2m) equivalent expressions?

Show why or why not.

Solve ItYou can combine like terms, then expand.

2 1 ·· 3 (23m 1 6 2 12 1 15m)

2 1 ·· 3 (12m 2 6)

2 1 ·· 3 (12m) 2 1 2 1 ·· 3 2 (6)

24m 1 2

2(1 2 2m)

2(1) 2 2(2m)

2 2 4m

24m 1 2

Solve ItYou can expand, then combine like terms.

2 1 ·· 3 (23m 1 6 2 12 1 15m)

2 1 ·· 3 (23m) 1 1 2 1 ·· 3 2 (6) 2 1 2 1 ·· 3 2 (12) 1 1 2 1 ·· 3 2 (15m)

m 2 2 1 4 2 5m

24m 1 2

2(1 2 2m)

2(1) 2 2(2m)

2 2 4m

24m 1 2



CONNECT IT

➤➤ Use the problem from the previous page to help you understand how to find equivalent expressions.

How do the Solve Its show that the expressions are equivalent?

How can expanding two expressions help show that the expressions are equivalent?

Is 2 1 ·· 3 (3 2 m) equivalent to 21 1 1 ·· 3 m or to 21 2 1 ·· 3 m? Explain.

How is expanding an expression with negative terms like expanding an expression with positive terms? How is it different?

Reflect Think about all the models and strategies you have discussed today. Describe how one of them helped you better understand how to determine whether expressions are equivalent.

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Apply It

➤➤ Use what you learned to solve these problems.

Raúl says the expression 225 1 1 ·· 5 x 2 20 2 is equivalent to 25x 2 500. Do you agree?

Explain why or why not.

Which expressions are equivalent to 24(225y 1 4 1 50y 2 8)? Select all that apply.

A 2100y 2 4

B 22(23y 2 5 1 27y 2 3)

C (100y 1 32) 2 (200y 1 16)

D 216 2 100y

E 2(2100y 1 16 1 200y 2 32)

F 2300y 2 48

Are the expressions 2(3x 2 6) and 2(3x) 2 6 equivalent? Explain your reasoning.

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7

8



Practice Expanding Expressions➤➤ Study the Example showing how to decide whether two expressions

are equivalent. Then solve problems 1–5.

Example

Is 2 3 ·· 4 (28a 2 12) equivalent to 6a 1 9?

You can start by expanding 2 3 ·· 4 (28a 2 12).

2 3 ·· 4 (28a 2 12)

2 3 ·· 4 [28a 1 (212)]

1 2 3 ·· 4 2 (28a) 1 1 2 3 ·· 4 2 (212)

6a 1 9

You can rewrite 2 3 ·· 4 (28a 2 12) as 6a 1 9. So, the two expressions

are equivalent.

Look at the Example. How do you know that 1 2 3 ·· 4 2 (28a) 1 1 2 3 ·· 4 2 (212) is

equivalent to both 2 3 ·· 4 (28a 2 12) and 6a 1 9?

Bianca makes an error when she tries to write an expression equivalent to 12 1 15(3 2 y) 2 10y. What is the error? Fix Bianca’s error.

12 1 15(3 2 y) 2 10y

12 1 45 1 15y 2 10y

57 1 5y

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2


Vocabularyequivalent expressions two or more different expressions that always name the same value.


A square playground is surrounded by a sidewalk on all sides. The sidewalk is 2n 1 3 yd long on each side of the park. The sidewalk is 0.5 yd wide. Write two equivalent expressions for the perimeter of the playground. Show your work.

SOLUTION

Is 2 2 ·· 3 (212b 2 6 1 9b 2 18) equivalent to 2(b 1 8)? Show your work.

SOLUTION

Juanita says that 3.5[4d 2 (2)(1.5)] and 2[7d 2 (5)(1.05)] are equivalent. Is Juanita correct? Explain your reasoning.

3

4

5


0.5 yd2n + 3 yd



Are the expressions 24c 2 36 1 12 2 12c 1 36 2 48c and 22(18c 2 6) equivalent? Show why or why not.

Develop Factoring Expressions


TRY IT Math Toolkit algebra tiles, grid paper

Ask: How did you reach that conclusion?

Share: First, I . . .

DISCUSS IT


➤➤ Explore different ways to determine whether expressions are equivalent.

Are the expressions 24c 2 36 1 12 2 12c 1 36 2 48c and 22(18c 2 6) equivalent? Show why or why not.

Solve ItYou can combine like terms and then expand the expression that has parentheses.

24c 2 36 1 12 2 12c 1 36 2 48c

(24c 2 12c 2 48c) 1 (236 1 12 1 36)

236c 1 12

22(18c 2 6)

236c 1 12

Solve ItYou can combine like terms and then factor.

24c 2 36 1 12 2 12c 1 36 2 48c

(24c 2 12c 2 48c) 1 (236 1 12 1 36)

236c 1 12

22(18c 2 6)

22(18c 2 6)

Solve ItYou can factor each expression before combining like terms.

24c 2 36 1 12 2 12c 1 36 2 48c

12(2c 2 3 1 1 2 c 1 3 2 4c)

12(23c 1 1)

22(18c 2 6)

22[26(23c 1 1)]

12(23c 1 1)



CONNECT IT

➤➤ Use the problem from the previous page to help you understand how to use the distributive property to find equivalent expressions.

How does each of the Solve Its show that the expressions are equivalent?

In the third Solve It, the second expression is already in factored form. Why can it be factored again?

How is factoring an expression with negative terms similar to factoring an expression with positive terms? How is it different?

Combining like terms, expanding, and factoring are strategies you can use to write equivalent expressions. When might you want to use each of these strategies?

Reflect Think about all the models and strategies you have discussed today. Describe how one of them helped you better understand how to determine whether expressions are equivalent.

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Apply It


A cell phone company is having a sale. The expression

12c 1 12 1 3 ·· 4 c 2 1 12 1 1 ·· 2 c 2 shows the total cost for buying 3 phones

that each cost c dollars per month for 12 months.

Which expressions are equivalent to that expression? Select all that apply.

A 12(2.25c)

B 12c 1 12 3 ·· 4 c 1 12 1 ·· 2 c

C 12c 1 8c 1 6c

D 37 1 ·· 4 c

E c(12 1 9 1 6)

Are (9g 2 11 1 10g) 2 (12 2 11g 1 13) and 23(210g 1 12) equivalent expressions? Explain.

Is 2(6 2 3x) 1 x equivalent to 2(3x) 1 x 1 12? Show your work.

SOLUTION

6

7

8


2nd Phone:25% o�

3rd Phone:50% o�

Cell Phone Sale Today!

1st Phone:Full Cost


Practice Factoring Expressions➤➤ Study the Example showing how to use factoring to write an equivalent

expression. Then solve problems 1–6.

Example

Consider the expression (23t 1 3) 1 (2 2 2t) 1 (24t 1 4). Write an equivalent expression that is the product of two factors.

You can combine like terms. Then find a common factor.

(23t 1 3) 1 (2 2 2t) 1 (24t 1 4)

(23t 2 2t 2 4t) 1 (3 1 2 1 4)

29t 1 9

29[t 1 (21)]

An equivalent expression is 29(t 2 1).

Write 29t 1 9 as the product of two factors in a way that is not shown in the Example. Explain how you found it.

Write an expression equivalent to 6 2 4(3 2 6m) 1 12m that is the product of two factors. Show your work.

SOLUTION

Is 1 1 4(3x 2 10) 2 12x equivalent to 239? Explain.

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Vocabularyequivalent expressions two or more different expressions that always name the same value.

factor (noun)a number, or expression within parentheses, that is multiplied.

factor (verb)to rewrite an expression as a product of factors.


Olivia and Isabella each earn d dollars for each dog she walks. One day, Olivia walks 6 dogs and gets $8 in tips. That same day, Isabella walks 9 dogs and gets $12 in tips.

a. Olivia writes 6d 1 8 1 9d 1 12 to represent the amount they earn together. Isabella writes 5(3d 1 4). Are their expressions equivalent? Show your work.

SOLUTION

b. Olivia and Isabella want to find out how much money they earn altogether. Suppose they earn $12 for each dog walk. Will you get a different amount if you evaluate Isabella’s expression instead of Olivia’s? Explain.

Are 1 ·· 2 x 1 3 ·· 4 2 5 ·· 8 x 2 7 ·· 8 and 1 ·· 8 (x 1 1) equivalent expressions? Show your work.

SOLUTION

Show that 20.75(24f 1 12) and (5f 1 9) 2 (2f 1 18) are equivalent expressions.

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5

6


6 walks

Dog Walking

Olivia Isabella

$8 in tips

9 walks

$12 in tips


➤➤ Complete the Example below. Then solve problems 1–9.

Example

Consider the expression 2 3 ·· 4 (24f 1 3g 1 8f 2 5g). Write an equivalent

expression that is the sum of two terms.

Look at how you could rewrite an expression with more than one variable.

2 3 ·· 4 (24f 1 3g 1 8f 2 5g)

2 3 ·· 4 (24f 1 8f 1 3g 2 5g)

2 3 ·· 4 (4f 2 2g)

SOLUTION

Apply It

Are 0.5 2 3(22x 2 1 ·· 3 1 4 1 8x) and 21.5(12x 1 7) equivalent expressions?

Show your work.

SOLUTION

CONSIDER THIS . . .To be like terms, the terms must have the same variables.

PAIR/SHAREWhat would happen if you factored 4 out of (4f 2 2g)?

1CONSIDER THIS . . .Sometimes it is easier to see common factors after you combine like terms.

PAIR/SHAREWhat is another way you could show that your answer is correct?

Refine Writing Equivalent Expressions Involving Rational Numbers




Are the expressions 8(9 2 6x 1 11) and 15 1 3 ·· 2 (232x 1 120) 2 35 both

equivalent to 216(3x 2 10)? Show your work.

SOLUTION

Which expression is equivalent to 23(10m 2 2) 1 (3 1 6m 2 3)?

A 224m 1 6

B 224m 2 6

C 248m 1 6

D 218m

Evelyn chose A as the correct answer. How might she have gotten that answer?

2CONSIDER THIS . . .You can start by expanding, factoring, or combining like terms.

PAIR/SHAREIs it possible for both expressions to be equivalent to 216(3x 2 10) but not to each other? Why or why not?

3CONSIDER THIS . . .The product of two negative numbers is positive.

PAIR/SHAREHow could you check your answer?


In front of a store, there is a row of parking spaces. Cars park parallel to one another, with the front of each car facing the store. Currently there are 10 compact spaces and 12 full size spaces. The store owners think they can repaint in the same space to fit 16 compact spaces and 9 full size spaces. The width of each type of space will not change. Are the store owners correct? Explain.

The variable z represents a positive integer. Does 4 1 3(2z 2 5) represent a number that is greater than, less than, or equal to 2(3z 2 4)? Show your work.

SOLUTION

Which expressions are equivalent to 1 ·· 5 x(5y 1 60)? Select all that apply.

A 1 ·· 5 (2xy 1 20x 1 3xy 1 40x)

B xy 1 60x

C y 1 12x

D 25xy 1 300x

E 13xy

F x(y 1 12)

4

5

6

FULL SIZE SPACE

COMPACT SPACE

w

1.125w

PARKING SPACE PLAN


Tell whether each statement is True or False.

True False

a. 6(25a 1 4) and 3(210a 1 8) are equivalent expressions.

b. 1 ·· 6 (15a 2 36) and 1 ·· 2 a 2 (3a 1 6) are equivalent expressions.

c. 23a 1 1.5 and 20.5(6a 1 1.5) are equivalent expressions.

d. 24(a 2 4 1 2a 1 8) and 24(a 2 4) 2 4(2a 1 8) are equivalent expressions.

Kazuko says the expressions 5x and 6 2 x are equivalent expressions, because you can substitute 1 for x in both expressions and get the same result. Is Kazuko’s reasoning correct? Explain.

Math Journal Start with the expression 12 2 3 ·· 4 1 8f 2 5 ·· 3 1 4f 1 2 ·· 3 2 . Write an

equivalent expression, and explain why it is equivalent. Then write an expression

that is not equivalent and explain why it is not equivalent.

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End of Lesson Checklist INTERACTIVE GLOSSARY Find the entry for factor (verb). Add two things you

learned about factoring in this lesson.

SELF CHECK Go back to the Unit 4 Opener and see what you can check off.

©Curriculum Associates, LLC Copying is not permitted. LESSON 18 Write and Solve Multi-Step Equations 355

Previously, you learned how to reason about equations to find unknown values. In this lesson, you will learn about solving equations algebraically.

➤➤ Use what you know to try to solve the problem below.

Adela, Rachel, and Santo take pictures at a Purim celebration. • Adela takes 7 more pictures than Rachel. • Santo takes 4 times as many pictures as Adela. • Santo takes 48 pictures. How many pictures does Rachel take?

Explore Solving Multi-Step Equations


TRY IT Math Toolkit algebra tiles, grid paper, number lines, sticky notes

Ask: What did you do first to find the number of pictures Rachel takes? Why?

Share: I started by . . . because . . .

DISCUSS IT

Learning Targets SMP 1, SMP 2, SMP 3, SMP 4, SMP 5, SMP 6, SMP 7Use variables to represent quantities and construct simple equations to solve problems.• Solve word problems leading to equations of the form px 1 q 5 r and p(x 1 q) 5 r, where p, q, and r

are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.

©Curriculum Associates, LLC Copying is not permitted.LESSON 18 Write and Solve Multi-Step Equations356

CONNECT IT

Look Back How many pictures do Adela and Rachel each take? How do you know?

Look Ahead One way to find the number of photos Adela and Rachel each take is to reason about the quantities arithmetically. Another way is to solve an equation algebraically. Look at two ways you could find the unknown in the statement the product of 6 and a number, n, plus 4 is 22.

Arithmetic Approach

Think: What number is 4 less than 22?

Step 1: 22 2 4 5 18

Think: What number times 6 is 18?

Step 2: 18 4 6 5 3

The number is 3.

Algebraic Approach

6n 1 4 5 22

Step 1: 6n 1 4 2 4 5 22 2 4

6n 5 18

Step 2: 6n 4 6 5 18 4 6

n 5 3

a. How is Step 1 in the arithmetic approach like Step 1 in the algebraic approach?

b. How is Step 2 in the arithmetic approach like Step 2 in the algebraic approach?

c. Why do both approaches lead to the same solution?

Reflect How is the algebraic approach similar to the arithmetic approach? How is it different?

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Rosa says 4x and 26x are like terms, so they can be combined. Tiffany says 5a and 5b are like terms, so they can be combined. Is Rosa correct? Is Tiffany correct? Why or why not?

2

Prepare for Writing and Solving Multi-Step Equations

Think about what you know about the like terms in an expression. Fill in each box. Use words, numbers, and pictures. Show as many ideas as you can.

1


What Is It?

Examples

What I Know About It

Non-Examples

like terms


Kaley, Safara, and Daniel keep track of how many graphic novels they read over the summer. • Kaley reads 6 graphic novels fewer than Safara. • Daniel reads 3 times as many as Kaley. • Daniel reads 30 graphic novels.

a. How many graphic novels does Safara read? Show your work.

SOLUTION

b. Check your answer to problem 3a. Show your work.

3


Daniel

Reads 3 timesas many

graphic novelsas Kaley

Reads 6 fewergraphic novels

than Safara

Reads ?graphic novels

Kaley Safara



Noah is designing a set for a school theater production. He has 150 cardboard bricks. He needs to use some of the bricks to make a chimney and 4 times as many bricks to make an arch. He also saves 15 bricks in case some get crushed. How many cardboard bricks can he use to make the arch?

Develop Writing and Solving Equations With Two or More Addends


TRY IT Math Toolkit algebra tiles, grid paper, number lines, sticky notes

Ask: How would you explain what the problem is asking in your own words?

Share: The problem is asking . . .

DISCUSS IT

LESSON 18 Write and Solve Multi-Step Equations360


➤➤ Explore different ways to find an unknown value in an equation that has two or more addends.

Noah is designing a set for a school theater production. He has 150 cardboard bricks. He needs to use some of the bricks to make a chimney and 4 times as many bricks to make an arch. He also saves 15 bricks in case some get crushed. How many cardboard bricks can he use to make the arch?

Model ItYou can draw a bar model to make sense of the problem.

Let x represent the number of bricks in the chimney.

150

x 4x 15

Use the model to write an equation.

x 1 4x 1 15 5 150

5x 1 15 5 150

Model ItYou can start solving the equation by isolating the x-term.

5x 1 15 5 150

5x 1 15 2 15 5 150 2 15

5x 5 135

Model ItYou can start solving the equation by dividing both sides by the same value.

5x 1 15 5 150

(5x 1 15) ······· 5 5 150 ··· 5

x 1 3 5 30

©Curriculum Associates, LLC Copying is not permitted.


CONNECT IT

➤➤ Use the problem from the previous page to help you understand how to solve an equation that has two or more addends.

How many bricks can Noah use to make the arch?

Look at the first Model It. How does the bar model represent the situation?

Look at the second Model It. Why do you subtract 15 from both sides? What do you need to do next to find the value of x?

Look at the third Model It. Why do you divide all of the terms by 5?

Look at the second and third Model Its. How are the strategies for solving 5x 1 15 5 150 similar? How are they different?

Describe two ways you could solve the equation 2x 1 12 5 8.

Reflect Think about all the models and strategies you have discussed today. Describe how one of them helped you better understand how to solve the Try It problem.

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Apply It


Solve 221 5 2 1 ·· 4 y 1 6. Show your work.

SOLUTION

A rectangular garden sits next to a house. There is fencing on three sides of the garden and the fourth side is the house. There is a total of 21.5 meters of fencing around the garden. The length of the garden along the house is 9 meters. Which equation can be used to find the width, w, of the garden in meters?

A 2w 1 9 5 21.5

B 2w 1 18 5 21.5

C 2w 2 21.5 5 9

D 2w 1 21.5 5 9

The total cost of a sketchpad and 6 pencils is $22.53. The sketchpad costs $9.99. Each pencil costs the same amount. How much does each pencil cost? Show your work.

SOLUTION

8

9

10


Practice Writing and Solving Equations With Two or More Addends

➤➤ Study the Example showing how to solve a problem using an equation. Then solve problems 1–5.

Example

Chloe is making a mural. She spends 6 hours designing it. She paints it during 3 sessions. Each session is the same number of hours long. In all, Chloe spends 24 hours making the mural. How many hours long, h, is each painting session?

You can represent the situation with an equation.

3h 1 6 5 24

3h 1 6 ······ 3 5 24 ·· 3

h 1 2 5 8

h 1 2 2 2 5 8 2 2

h 5 6

Each painting session is 6 hours long.


Demarco has a piece of fabric 6 yd long. He

uses a piece 3 yd long. He cuts the rest into

strips that are each 3 ·· 4 yd long. How many

3 ·· 4 yd long strips are there? Show your work.

SOLUTION

1 Solve 27 5 12x 2 16. Show your work.

SOLUTION

2


Liam makes soap sculptures of sea turtles. Each sculpture

weighs 3 ·· 8 pound. He ships them in a wooden box that weighs

2 pounds. The total weight of the box filled with the t sea turtles

is 5 pounds. How many sea turtles are in the box? Show

your work.

SOLUTION

Solve 20.4k 2 6 5 1.2. Show your work.

SOLUTION

Claudia buys 12 postcards, 12 stamps, and 1 pen. The postcards cost twice as much as the stamps. The pen costs $1.50. The total cost is $14.10. How much does each postcard cost? Show your work.

SOLUTION

3

4

5




Hugo is traveling in Toronto, Canada. His weather app shows the

temperature is 25°C. Hugo writes the equation 25 5 5 ·· 9 (F 2 32)

to find the temperature in degrees Fahrenheit, F. What is the

temperature in degrees Fahrenheit?

Develop Writing and Solving Equations with Grouping Symbols


TRY IT Math Toolkit grid paper, number lines, sticky notes

Ask: Why did you choose that strategy to find the temperature in degrees Fahrenheit?

Share: I knew . . . so I . . .

DISCUSS IT

1:00 PMCarrier 100%

TorontoCurrent Temperature

25°CNOW 3PM 4PM 5PM 6PM 7PM

25°C 25°C 25°C 24°C 23°C 23°C

Wednesday

Thursday

Friday

Saturday

25°C

27°C

25°C

27°C

➤➤ Explore different ways to find an unknown value in an equation with grouping symbols.

Hugo is traveling in Toronto, Canada. His weather app shows the temperature is

25°C. Hugo writes the equation 25 5 5 ·· 9 (F 2 32) to find the temperature in

degrees Fahrenheit, F. What is the temperature in degrees Fahrenheit?

Model ItYou can use the distributive property to expand.

25 5 5 ·· 9 (F 2 32)

25 5 5 ·· 9 (F) 2 5 ·· 9 (32)

25 5 5 ·· 9 F 2 160 ··· 9

25 1 160 ···· 9 5 5 ·· 9 F 2 160 ··· 9 1 160 ···· 9

385 ··· 9 5 5 ·· 9 F

Model ItYou can divide each side by the coefficient 5 ·· 9 .

25 5 5 ·· 9 (F 2 32)

25 4 5 ·· 9 5 5 ·· 9 (F 2 32) 4 5 ·· 9

25 • 9 ·· 5 5 5 ·· 9 (F 2 32) • 9 ·· 5

45 5 F 2 32




CONNECT IT

➤➤ Use the problem from the previous page to help you understand how to solve an equation with grouping symbols.

What is 25°C in degrees Fahrenheit?

Look at the first Model It. Describe the steps shown for solving the equation. What do you still need to do to find the value of F?

Look at the second Model It. Describe the steps shown for solving the equation. What do you still need to do to find the value of F?

Look at the Model Its. What was one advantage of distributing first? What was one advantage of dividing first?

Consider the equation 12 5 b(2.5x 1 15). What values of b might make you want to start solving the equation by distributing b? What values of b might make you want to start solving the equation by dividing by b?

Reflect Think about all the models and strategies you have discussed today. Describe how one of them helped you better understand how to write and solve an equation that includes grouping symbols.

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Apply It


Carolina fosters 5 puppies. For each puppy she buys a crate that costs c dollars and a leash that costs $20. She spends $475 total. Which equations model the situation? Select all that apply.

A 5c 1 20c 5 475 B 5(c 1 20) 5 475

C 5c 1 100 5 475 D 5c 1 20 5 475

E c 1 20 5 475

Solve 28 5 k 2 4 ····· 26 . Show your work.

SOLUTION

The perimeter of a rectangular chicken coop is 30 feet. The width is w feet and the length is w 1 4 feet. What are the length and width of the coop? Show your work.

SOLUTION

7

8

9

w

w + 4


Practice Writing and Solving Equations with Grouping Symbols

➤➤ Study the Example showing how to use an equation with grouping symbols to solve a problem. Then solve problems 1–5.

Example

Lillie and her family donate money to charity at the end of each year. Lillie’s brother donates $3 more than Lillie. Her parents donate 4.5 times as much as Lillie’s brother. Lillie’s parents donate $45. How much does Lillie donate?

You can represent the situation with an equation.

d 5 Lillie’s donation in dollars

4.5(d 1 3) 5 45

4.5(d 1 3) ········ 4.5 5 45 ··· 4.5

d 1 3 5 10

d 1 3 2 3 5 10 2 3

d 5 7

Lillie donates $7.


Look at 4.5(d 1 3) 5 45 from the Example.

a. What does (d 1 3) represent?

b. Why is (d 1 3) multiplied by 4.5?

c. How much does Lillie’s brother donate?

1 Malik joins a gym. He gets $2 per month off the regular monthly rate for 3 months. Malik pays $49.50 for 3 months. What is the gym’s regular monthly rate, r? Show your work.

SOLUTION

2



Luis is shopping for gifts. Mugs are on sale for $4 off the regular price, p. Luis buys 6 mugs. He pays a total of $54. What is the regular price of a mug? Show your work.

SOLUTION

Solve 3 ·· 4 (5x 2 3) 1 8 5 17. Show your work.

SOLUTION

Solve 272 5 8(y 2 3). Show your work.

SOLUTION

3

4

5


➤➤ Complete the Example below. Then solve problems 1–8.

Example

Solve 20.25x 1 7.5 5 15.

Look at how you could show your work using multiplication.

20.25x 1 7.5 5 15 100(20.25x 1 7.5) 5 (100)15 225x 1 750 5 1,500 225x 1 750 2 750 5 1,500 2 750 225x 5 750

SOLUTION

Apply It

Solve 0 5 21.8y 1 0.72. Show your work.

SOLUTION

CONSIDER THIS . . .You can multiply both sides by a power of 10 to eliminate the decimals.

PAIR/SHAREWhat is another way you could solve this problem?

1CONSIDER THIS . . .You can think of 0 5 21.8y 1 0.72 as having two addends.

PAIR/SHAREHow can you check your answer?

Refine Writing and Solving Multi-Step Equations



Solve 2(n 1 17) ········ 8 5 3 ·· 8 . Show your work.

SOLUTION

Three siblings are born on the same date in consecutive years. The sum of their ages is 42. What is the age of the oldest sibling?

A 13

B 14

C 15

D 16

Victoria chose A as the correct answer. How might she have gotten that answer?

2CONSIDER THIS . . .There is more than one way to think about this problem.

PAIR/SHAREHow did you choose your first step?

3CONSIDER THIS . . .Consecutive integers follow each other, like 4, 5, 6. If the first integer is x, the next is x 1 1, then x 1 2, and so on.

PAIR/SHAREHow would the answer change if there were four siblings?



Leon pays $12.50 per month for a music subscription service. One month he also buys 6 songs from the service. Each song costs the same. His bill for that month is $17.84. In dollars, how much does he pay for each song?

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0 0 0 0 0 0

One side of an isosceles triangle is 2x 1 1 ft long. The other two sides are both 3x 2 1 ft long. The perimeter of the triangle is 55 ft. What is the length of each side? Show your work.

SOLUTION

Khalid is solving the equation 8.5 2 1.2y 5 6.7. He gets to 1.8 5 1.2y. Explain what he might have done to get to this equation.

4

5

6

1:00 PMCarrier 100%This Month’s Purchases

6 Songs

I Love You To The Moon And Back

1:04 3:28

Carrier 10:20 AM 100%



Mora preparing her pack for a hike. Her empty pack weighs 15 ·· 16 pound. She

adds some water bottles that each weigh 1 1 ·· 8 pound. Now Mora’s pack weighs

6 9 ·· 16 pounds. How many bottles, b, does Mora add to her pack? Show your work.

SOLUTION

Solve 1 ·· 2 1 1 ·· 3 w 5 1 ·· 6 . Show your work.

SOLUTION

Math Journal Damita says the equations 0.8x 2 0.8 5 1.6 and 4 ·· 5 (x 2 1) 5 1 3 ·· 5

are the same. How can she show this, without solving the equations?

7

8

9

End of Lesson Checklist INTERACTIVE GLOSSARY Write a new entry for represent. Write at least one

synonym for represent.

SELF CHECK Go back to the Unit 4 Opener and see what you can check off.

explore equivalent expressions · involving rational numbers think about what you know about the...

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